So now you can write down a function that, given the coordinates of the center of the circle and the radius, and the value for x it returns the coordinate y and use these coordinates to move the turtle:

By choosing one of the two coordinates returned you choose whether to draw the "upper" or "lower" arc.

to draw a smile you simply have to come up with two circles one smaller and a larger one but with the center slightly above the previous one so that they have that kind of intersection.

So you have to choose a circle C1 centered in x0, y0 with radius r and a circle C2 centered in x0, y0+K with radius R > r.
Note that C2's center is vertically aligned with C1 center (hence the same x coordinate for the center) but it is above it (note: I'm not sure of y-axis orientation so the +K might be -K...)

To find the intersections you have to solve the system of their equations: